Small lightweight aircraft navigation in the presence of wind

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Small lightweight aircraft navigation in the presence of wind

Cornel-Alexandru Brezoescu

To cite this version:

Cornel-Alexandru Brezoescu. Small lightweight aircraft navigation in the presence of wind. Other.

Université de Technologie de Compiègne, 2013. English. �NNT : 2013COMP2105�. �tel-01060415�

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Par Cornel-Alexandru BREZOESCU

Thèse présentée

pour l’obtention du grade de Docteur de l’UTC

Navigation d’un avion miniature de surveillance aérienne en présence de vent

Soutenue le 28 octobre 2013

Spécialité : Laboratoire HEUDIASYC

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Navigation d’un avion miniature de surveillance a´ erienne en pr´ esence de vent

Student: BREZOESCU Cornel Alexandru

PHD advisors : LOZANO Rogelio

CASTILLO Pedro

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Chapter 1 Introduction

Performance improvement is a requirement for all physical systems including Un- manned Aerial Vehicles (UAVs) whose applications are steadily increasing. They offer a smooth transition of autonomous flight control design from theory to practice in addition to providing a proper solution in environments inaccessible or dangerous to human life. However, the lack of a human pilot on board implies that the UAVs rely on automation to navigate or to avoid obstacles. Thus, given the complex dynamics of a flying vehicle, a flight controller capable to provide, maintain and improve the aircraft performance is required in order to guarantee the stability of the system despite uncertainties in the model or some external perturbing forces like wind.

1.1 Motivation and objectives

This work is motivated by commonly encountered situations which are dangerous to human life such as extreme manifestations of force or hostage crisis, among others.

The number of casualties would be significantly reduced if information on the site were

available before the procedure of the emergency services. This information could be

obtained by the use of a drone which must be able to navigate a safe distance to the

place of intervention and back to the ground station in an unobtrusive way. However,

the relatively low operating speed of small UAVs makes them particularly affected by

wind field which is any movement of the air mass with respect to the surface of the

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Earth. Therefore, the drone in question must also be able to overcome the effect of such perturbation in order to safely meet the mission objectives.

Miniature drones seem to be best suited to solve this kind of problems due to their discretion and portability. Nevertheless, it is necessary to increase their flight time and make them more tolerant to wind. Note that the wind may be of the order of the drone speed, which makes the problem very difficult.

Based on these considerations, the aim of this thesis is to develop a navigation strategy for a model aircraft of conventional configuration in order to accomplish different missions in presence of wind. If it is possible to obtain the path that minimizes the time of flight, the energy consumption or the forces acting on the structure of the vehicle, then the flight controller must be able to steer the vehicle along this path. As for what concerns the direction and intensity of the wind, the information provided by a ground station can be taken into account to define the trajectory of the mission.

Since the aircraft is considered small, wind has a significant influence on its flight performance. Note that the wind measured by the ground station may be different from the one actually encountered at higher altitudes or at any location away from the initial point of measurement. To improve the performance of flight, a method for estimating the intensity and the direction of the wind will be explored and the navigation strategy will be adapted accordingly. To this end, the measures provided by the ground station and the changes in the aircraft trajectory due to the wind must be considered. To simplify the study, a first assumption will be to consider that the wind gusts are isolated. Thus, the wind field is relatively constant or varies slowly over time. Finally, the proposed navigation strategies need to be validated in real flight tests using an experimental prototype which also must be developed.

1.2 Challenges

The large flight envelope of the aerial devices posses several challenges to the suc-

cessful achievement of the proposed objectives. First, the choice of a robust airframe

possessing reliable flight characteristics is essential for real flight tests. Long duration

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flight and sufficient payload capacity to carry the weight of sensors and batteries are two features of great interest. Then, the appropriate avionics equipment meeting the requirements of the application being treated in this work needs to be integrated.

Onboard sensors are required, on the one hand, to provide measurements of the quantities needed to determine the relevant parameters of the nonlinear dynamic model of the airplane. For this purpose, the use of parameter identification techniques is mandatory. On the other hand, the avionics system must provide the parameters of any integrated or developed flight controller.

This emphasizes the second main challenge that must be faced when dealing with UAVs, which is the development of a control law capable of governing the airplane during the flight. The motion of an airplane through the air is modeled as a coupled nonlinear system with complex dynamics. Its performance depends on the operating altitude, speed, atmosphere or the geometric characteristics of the airplane. That involves an increased complexity in the implementation of flight controllers which should provide both the internal stability of the aircraft and the achievement of the commanded attitude and velocity. In addition, the controller must be able to remove the adverse effects of external disturbances or model uncertainties in order to maintain good performance.

Last but not least, developing autonomous aerial devices requires a good understanding of the principles of aircraft operation. With this stock of knowledge, the influence of wind on aircraft performance can be incorporated into the equations of motion and, then, compensated with appropriate control input.

1.3 Approach

In order to achieve these objectives, the following research areas have been addressed.

First, a literature review on the subject is necessary in order to obtain the aerody-

namic model of the miniature aircraft in the presence of wind. Secondly, the devel-

opment of navigation strategies is required in order to allow the airplane to perform

various tasks in the presence of wind. To improve the performance of flight, the nav-

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igation strategies need to take into account the wind measured by a ground station or estimated by an online estimator based on the aerodynamic model.

The third axis is the design and implementation of an experimental setup which consists of a ground station used for visualization and control purposes and an em- bedded autopilot architecture containing the airframe platform equipped with appropriate avionics such as inertial measurement unit, global positioning system, commu- nication devices, air sensors and a control processing unit to manage the control law and the sensors data.

1.4 Thesis outline

The manuscript is divided in two main sections which are: modeling of the aircraft dynamics and designing flight controllers in order to achieve autonomous flight in presence of wind. The first section focuses on deriving complete and reduced-order mathematical models for a fixed-wing UAV and it is represented by Chapter 2. In the first part of the section, a description of the physical principles of flight is presented with the aim of deriving the dynamic model of an airplane. The additional forces acting on the aircraft subjected to wind, which is modeled as a stochastic process, have been incorporated into the equations of motion. Further, this section presents a reduced-order model of the airplane which is appropriate for control design.

On the other hand, the second aspect addressed in this report, i.e., achieving au-

tonomous flight in presence of wind, is described in chapters 3 and 4. The control

design developed in these chapters relies on the simplified model introduced in the

previous section of the thesis. Furthermore, chapter 5 describes the practical imple-

mentation of a test platform with the purpose of obtaining flight test results. Finally,

general conclusions are presented in chapter 6.

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Chapter 2 Modeling for control

To address the problem of designing an autonomous flight controller for a small fixed- wing UAV, first an accurate nonlinear dynamic model of the vehicle needs to be derived. Unlike ground transportation systems, whose motion is primarily governed by propulsive inputs, airplanes rely on aerodynamic forces which are difficult to model since they depend on many varying operating conditions. As a consequence, the development of autonomous operating aerial devices is a challenging problem which requires significant attention.

The present chapter begins by introducing the basic principles of flight and the common parts of an airframe. Then, the derivation of the airplane equations of motion is described in order to formulate the problem from an automatic control perspective.

Finally, the effect of a moving atmosphere on the aircraft performance is discussed and the wind is incorporated into the mathematical model of the vehicle.

2.1 Basic principles of flight

The design of an effective flight controller for an autonomous UAV starts with a good

understanding of the principles of flight theory. A lack of knowledge about basic

aerodynamics may cause inappropriate input commands when the aircraft operates

at the limit of its performance capabilities. For this reason, the objective of this

section is to provide a basic insight into the mechanics of flight.

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2.1.1 The forces of flight

Fundamental aerodynamic principles involve the interaction between a solid object and the air which flows around the body of the object maintaining contact at all points. Considering the case of an aircraft in flight, the pressure variations along its component parts, caused by the physical contact with the air, generate an aerodynamic force which act through the center of pressure

¹

. This force can be resolved into a component normal to the airflow direction which is called lift, L, and a component along the airflow direction which is called drag, D.

L is always an upward force perpendicular to the flight direction and it depends on several variables. Likewise, D is a backward force highly sensitive to many factors and its main source is the skin friction between the air and the surface of the aircraft [36]. However, the theory explaining the generation of lift is more complex than the one justifying the drag and this difficulty has led to several incorrect descriptions which will be presented further in this chapter. There are two other forces acting on the airplane: thrust which is generated by the engines and which makes the aircraft to move forward and the gravitational force which is due to the weight of the airplane and is always directed downward the center of the Earth.

Consequently, the lift force is what holds the airplane in the air overcoming its weight while the thrust force is what moves the airplane forward overcoming drag.

When the airplane flies straight and level without accelerating, the four forces are in balance, thrust equalling drag and lift equalling weight. This particular case is represented in Figure 2 − 1.

The four forces affecting the flight of an airplane are vector quantities which means that they have both magnitude and direction. The motion of flight depends exclusively on the parameters of these vectors and on how they are related. In order for a pilot to manoeuvre the aircraft, the four forces have to be precisely manipulated.

Therefore, understanding their nature and possessing means to adjust their direction and magnitude is required in order to achieve precise control of the airplane.

1The center of pressure is the average location of all the pressure forces acting on the aircraft [31]. Similarly, the center of gravity is the average location of the weight of the airplane.

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Lift

Thrust Drag

Weight

Figure 2-1: The forces of flight

2.1.2 Parts of an airplane

Aircrafts may come in several configurations and sizes but they all work on the same principle, namely manipulating the for forces of flight. Therefore, any vehicle which is capable to provide these forces is said to be an aircraft, regardless its shape. But, since the subject of this thesis is fixed-wing UAVs, let us introduce the component parts of an aircraft of conventional shape.

An airplane consists of a propulsion system and many aerodynamic shapes which

can be fixed or variable. The propulsion system or the engine is used to power the

vehicle. The fixed aerodynamic shapes provide the lift force and the stability of

the airplane and they are represented by: fuselage, wings and tail stabilizers. With

respect to the variable aerodynamic shapes, they are commonly known as control

surfaces and they are divided in elevator, ailerons and rudder, see Figure 2 − 2. In

some aircrafts there are additional parts to vary the aerodynamics of the wing, most

of them being high lift devices such as flaps and slats. In addition, spoilers can be

employed to break the airflow over the wing or winglets to reduce drag.

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The fixed component parts of an airplane are listed below:

1. The propulsion system is the component which generates the thrust force required to move the airplane forwards. Both propeller and jet engines produce thrust by throwing the air backwards. By manipulating the power of the engine, one can control the magnitude of the resulting force while its direction is fixed along the longitudinal axis of the airplane.

2. The fuselage is the airplane component which connects all the parts together.

It has an aerodynamic shape in order to reduce the resulting drag force. Besides, a small proportion of produced lift comes from the fuselage.

3. The wings produce the most significant amount of lift which is the force that makes the flight of heavier-than-air vehicles possible.

4. Horizontal stabilizer is a small horizontal wing located at the tail of the airplane used to avoid up and down undesirable motion.

5. Vertical stabilizer - is a small vertical wing located at the tail of the airplane used to avoid side to side motion.

Engine Fuselage

Wings

Ailerons

Horizontal tail Vertical tail

Elevator Rudder

Figure 2-2: Component parts of an airplane

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The aerodynamic control surfaces are moving parts of the wings (including the tail stabilizers) that can change the airflow over these particular locations of the aircraft in very specific ways. Actually, they act by modifying the shapes of the wings and, thus, their cambers. This results in a desired pressure difference producing a controlled force. The common control surfaces of a fixed-wing aircraft are:

1. The elevator is a hinged-surface connected to the horizontal stabilizer which is used to control the vertical motion of the airplane. When the elevator is deflected downwards, the horizontal tail wing produces an increased lift force which makes the tail of the airplane to rise relative to the nose and, thereby, the airplane to descend

²

. An upward deflection of the elevator creates an opposite effect, making the aircraft to climb. Hence, the elevator controls the motion of the airplane around the lateral axis that is known as pitching motion.

2. The ailerons are movable sections placed outboard toward the wing tips which work usually in opposition: one deflected upward and one deflected downward.

They work in the same way as the elevator. As they are deflected, the airfoils chambers vary resulting an increased lift on one wing and a decreased lift on the other. The resulting motion of the airplane is a rotation around its longitudinal axis known as rolling.

3. The rudder is a variable part placed at the rear of the vertical stabilizer which causes the airplane to move from side to side. Deflecting the rudder, one can manipulate the amount of force produced by the vertical tail wing and, thereby, the motion of the aircraft around the vertical axis known as yawing.

Unlike pitching motion, rolling and yawing motions are not pure, that is rudder and ailerons deflections excite both yawing and rolling displacements. When rolling an airplane, the lowered aileron has more drag than the up-going aileron and this causes an adverse yaw. Therefore, the rudder is mainly used is to maintain the nose of the aircraft into the direction of flight, thus to obtain a coordinated flight.

2The aircraft descends when the elevator is deflected downward as a result of changing the angle of attack and the direction of the thrust vector.

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2.1.3 Misleading lift theories

At this point we can make a legit answer to the question ”How the airplanes fly?” by saying that there is a lift force produced by the wings which keeps the aircraft into the air. Such a general explanation satisfies just the curious but those ones really passionate about the flight of an airplane need a more detailed description of how this force is created. This is, for that matter, a much more difficult question which has been answered throughout time in several ways, but many concepts describing the basic principles of lift have been shown to be misleading and incorrect [32], such as ”the equal transit times” or ”the skipping stone” principles. Unfortunately, such misconceptions about flight have been taught for many years in most flight training manuals and they still create passionate debates between physicists and aeronautical engineers [38]. This subsection stars by examining the components of a wing in order to explain further some classical descriptions of lift.

Wing section

Let us first illustrate a wing section, usually called airfoil, as shown in Figure 2- 3. Notice from this figure that wings have generally a rounded leading edge and a pointed trailing edge. The line joining the center of the leading edge to the point of the trailing edge is called the chord line. When the wing has a curvature we speak about the camber of the airfoil. The airflow striking the aircraft and its component parts is called the relative wind and its direction is always opposite to and parallel with the flight path of the airplane.

Leading edge

Trailing edge Chord line

Relative wind

α

Angle of attack Flight path

Figure 2-3: Parts of a wing

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The relative wind and the flight path are horizontal and parallel in level flight.

However, the chord line of the airfoil forms a small angle, called the angle of attack α, with the flight path even in level flight. As it will be later discussed, α is an important variable in the flight dynamics since even small variations affect the amount of lift.

The equal transit theory

This subsection discusses one of the most popular incorrect explanations of the lift force. Although the theory is correct in principle, it fails to provide a satisfactory explanation in detail. The theory starts from a basic principle of aerodynamics proposed by the Swiss scientist Daniel Bernoulli who claims that the faster the air moves, the less pressure it exerts. In order to invoke this principle, the proponents of the theory state that the molecules of the air on the upper surface of the airfoil have to reach the trailing edge at the same time as the molecules on the lower surface [33]. Then, claiming that the top of the wing is shaped in order to provide a longer surface than the bottom, it follows that the air molecules have to generate higher velocities over the wing than underneath it. This difference in velocity is balanced by an increased air pressure under the wing which lifts the airplane into the air.

Figure 2−4 illustrates the ”equal transit time” theory. In this figure, the airfoil has a particular shape with the upper surface longer than the bottom. The air molecules split apart at the leading edge of the wing (point A) and they have to move faster over the top of the airfoil in order to meet at the trailing edge (point C). This difference in velocity produces a higher pressure underneath the wing and, thereby, lift.

Longer path

Shorter path

A B C

Figure 2-4: The ”equal transit time” lift theory.

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According to this theory, wings possessing a symmetrical airfoil have equal pressure on the both sides of the airfoil when flying in level flight, thus they do not produce any lift. But how do paper airplanes with perfectly flat wings fly? Or how can airplanes fly upside down since, using the same explanation based on Bernoulli principle, would show that the aircraft is pushed down. Moreover, how do the air particles know to speed up over the wing since they do not have any information about the geometry of the object with which they will interact? In reality, the air molecules on the upper surface of a wing travels at much higher velocity than the one required by the equal transit time theory. What goes wrong with this explanation is the fact that it uses the Bernoulli’s equation for the wrong assumption that the air molecules have to meet at the end of the wing. In addition, this theory fails to explain why the air moves faster over the wing than beneath it. Therefore, although Bernoulli’s argument is correct, the complete explanation is misleading.

The skipping stone theory

This theory uses the Newton’s laws of motion to explain the generation of lift. For this purpose, the wing is described as a surface which forces the air to go down.

Then, by Newton’s third law which states that ”for every action there is an equal and opposite reaction”, the lift is considered to be the reaction force of the airfoil to the air molecules striking the bottom surface of the wing. The name of the theory comes from the similarity with skipping a flat rock across a body of water when thrown at a small angle and its principle is shown in Figure 2 − 5.

Lift

Inflow

Outflow

Figure 2-5: The ”skipping stone” lift theory.

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The deficiency of this theory is that it does not consider the upper surface of the wing assuming that the downwash is all produced by the lower surface. Following this idea, it results that two airfoils having identical lower surfaces but different upper surfaces give the same amount of lift. In reality this is not the case, and this is due to the fact that the upper surface of the airfoil contributes more to the downwash than the lower surface.

2.1.4 Lift generated by airflow deflection

Lift is a mechanical force generated the by the airfoil of the airplane as it interacts with the air [35]. Indeed, the airfoil has an aerodynamic shape which produces a net deflection of the incoming flow since the molecules of the air stay in contact with the body of the wing. Hence, the airflow velocity vector is changed producing an acceleration. Finally, from Newton’s second law of motion, when a mass is accelerated then a force is produced.

Therefore, the wing creates lift as a reaction force from redirecting air downwards with the major part coming from the upper wing surface pushing air down [39]. Figure 2 − 6 shows the streamlines over a wing with lift generated by using the FoilSim III Java Applet provided by NASA. Notice from this figure that both the flow above and below the wing are bent down. In addition, the air passing above the wing travels faster than the air on the lower surface.

Figure 2-6: Lift generated by the airflow deflection.

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2.2 Coordinate frames

When analyzing the dynamics of an aircraft it is necessary to express its position and orientation relative to a suitable coordinate system in which the Newton’s laws of motion can be applied. In addition, several frames of reference need to be used in order to define relative positions and velocities, the choice of the frame to be employed being a matter of convenience. For example, sensors such as GPS measure the aircraft velocity relative to the Earth, thus a coordinate system fixed on Earth surface is preferable to write the velocity equations. On the other hand, sensors such as rate gyros give information with respect to the body of the vehicle where they are installed. Accordingly, the airplane angular rates are most easily described in a vehicle-fixed reference frame.

Based on these considerations, this section discusses the commonly used coordinate frames for the problem of the airplane flight dynamics and introduces the required transformations to bring vectors from one frame to another. The presentation in this section is mainly based on textbooks by B. Etkin [17] and Randal W.

Beard & Timothy W. McLain [40].

2.2.1 Inertial and Earth-fixed reference frames F

^I

, F

^E

Solving a dynamic problem requires an inertial reference frame, F

^I

, which is fixed or in uniform rectilinear translation relative to the distant stars. Meeting this requirement leads to the possibility of using the Newton’s second law for the motion of a particle, which relates the external forces acting on the particle to its mass and acceleration relative to F

^I

. Generally, the rotation of the Earth relative to such an inertial frame is neglected in the analysis of the flight dynamics. Therefore, any coordinate frame with the origin at a defined location on the Earth can be used as an inertial frame.

Let F

^E

denote an Earth-fixed frame having the origin close to the vehicle body

and its axes directed North, East and vertically down as shown in Figure 2 − 7. This

coordinate system will be used in further analysis to describe aircraft position and

orientation since many sensors measure these quantities with respect to the Earth.

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In addition, most mission are defined in this frame, such as waypoint guidance, flight trajectories etc.

F

I

x

i

y

i

z

i

North

East

Down

Figure 2-7: Illustration of the inertial coordinate frame.

2.2.2 Body-fixed coordinate system F

^B

The origin of the body-fixed frame, F

^B

, is identical to the vehicle center of gravity and the axes point out the nose of the airframe, out the right wing and downward, as shown in Figure 2 − 8. F

^B

has angular velocity relative to F

^I

denoted by ω = [p, q, r]

^T

. It is employed since the aerodynamic and propulsive forces act on the aircraft body and they are easily defined in this reference system. Moreover, on-board sensors generally measure information with respect to the body frame.

The orientation of F

^B

relative to F

^I

can be given by the Euler angles (ψ, θ, φ) which are three consecutive rotations about the axes z, y and x. The angles represent the yaw, pitch and roll and they rotate the body of the airplane about the vertical, lateral and longitudinal inertial axes. The transformations associated with each single rotation are given by

R

₁

(φ) =







1 0 0

0 cos φ sin φ 0 − sin φ cos φ







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R

₂

(θ) =







cos θ 0 − sin θ

0 1 0

sin θ 0 cos θ







R

3

(ψ) =







cos ψ sin ψ 0

− sin ψ cos ψ 0

0 0 1







The complete transformation from F

^I

to F

^B

reads R

_BI

= R

₁

(φ)R

₂

(θ)R

₃

(ψ)

=







cos θ cos ψ cos θ sin ψ − sin θ sin φ sin θ cos ψ sin φ sin θ sin ψ sin φ cos θ

− cos φ sin ψ + cos φ cos ψ

cos φ sin θ cos ψ cos φ sin θ sin ψ cos φ cos θ + sin φ sin ψ − sin φ cos ψ







(2.1)

2.2.3 Wind axes coordinate frame F

^W

The wind axes frame, F

^W

, has the origin at the aircraft center of gravity and the x-axis is directed along the velocity vector of the vehicle relative to the atmosphere as depicted in Figure 2 − 8. In calm conditions, i.e. atmosphere at rest, the origin of F

^W

will trace out the trajectory of the aircraft relative to the Earth.

The wind frame is of interest since the lift, drag and side forces are directly

measured in the direction of its axes. It has angular velocity relative to F

^I

and its

components are denoted by ω

_w

= [p

_w

, q

_w

, r

_w

]

^T

. The orientation of F

^W

relative to

the body-fixed frame is determined by the aerodynamic angles α and β which stand

for angle of attack and sideslip, respectively. This implies that some trigonometry is

required in order to bring the measured vectors from F

^W

into F

^B

or vice versa.

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xb

yb zb

β α

xw Va

Figure 2-8: Illustration of the body and wind axes reference frames.

In order to obtain the total transformation from wind to body frame, one may follow the same steps as for R

_BI

considering the sequence of rotations given by (−β, α, 0), where

R

2

(α) =







cos α 0 − sin α

0 1 0

sin α 0 cos α







R

₃

(−β) =







cos β − sin β 0 sin β cos β 0

0 0 1







Thus

R

BW

= R

1

(0)R

2

(α)R

3

(−β)

=







cos α cos β − cos α sin β − sin α

sin β cos β 0

sin α cos β − sin α sin β cos α







(2.2)

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2.3 Aircraft nonlinear model

The present section focuses on the derivation of the mathematical model of a fixed- wing UAV which is considered as a rigid body having 6 degrees of freedom. In addition, the Earth is considered as flat and stationary for the purpose of simplicity and that makes F

^E

a Newtonian frame of reference. Note that this is a valid ap- proximation for most problems of aircraft flight. The equations of motion are derived considering that the atmosphere is at rest relative to the Earth and making an appropriate adjustment at the end in order to include the effect of the wind into the model.

The presentation in this section is mainly based on the textbooks by [49, 17, 1].

2.3.1 State variables

The Earth relative aircraft motion can be described by position, orientation, velocity, and angular velocity over time. The position of the aircraft center of gravity in the inertial coordinate frame will be denoted by the vector p

_E

, whose components are

p

_E

= [p

_n

p

_e

− p

_h

]

^T

with p

n

being the inertial position along the North axis in F

^E

, p

e

representing the inertial position along the East axis in F

^E

and p

_h

denoting the inertial altitude along the vertical axis in F

^E

.

The Earth related aircraft orientation is represented by the Euler angles from the attitude vector Φ

Φ = [φ θ ψ]

^T

where φ is the roll angle, θ defines the pitch angle, and ψ represents the yaw angle.

The aircraft inertial velocity vector V

E

is commonly represented in several coordinate systems. Its components are given by

V

E

= R

EW

V

W

= R

EB

V

B

(2.3)

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where

V

_E

=





 v

_n

v

_e

v

_h







; V

_W

=





 V

_a

0 0







; V

_B

=





 v

_u

v

_v

v

_w







;

and R

_EW

is the transformation from F

^W

to F

^E

and it is obtained from the sequence of rotations given by the angles (φ

_w

, θ

_w

, ψ

_w

) which provide the orientation of the wind axes, V

_W

represents the inertial velocity vector measured in the direction of the wind axes reference frame, V

_a

denotes the magnitude of the aircraft velocity relative to the air mass known as airspeed, R

_EB

describes the rotation from F

^B

to F

^E

and R

_EB

= R

_IB

= R

^T_BI

while V

_B

represents the inertial velocity vector of the aircraft in the body coordinate system having components along the longitudinal, lateral and normal axes denoted by(v

_u

, v

_v

, v

_w

).

The velocity vector of the aircraft relative to surrounding air is denoted by V

_a

having the components V

a

= [u v w]

^T

in the body frame of reference. If the atmosphere is at rest, the air-relative aircraft velocity equals the velocity of the vehicle with respect to the Earth. Writing the equation describing this relationship for both body and wind axes, it follows

V

a

= V

B

= R

BW

V

W

and using equation (2.2), it yields





 u v w







= V

_a







cos α cos β sin β sin α cos β







which implies that

V

a

= √

u

²

+ v

²

+ w

²

α = arctan w

u β = arcsin v

V

_a

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The angular velocity vector is also represented in the body-fixed and in the wind- axes coordinate frames having the components

ω =





 p q r







; ω

_w

= R

_{W B}

ω =





 p

w

q

_w

r

w







where ω is the angular velocity vector of F

^B

relative to F

^E

, p denotes the roll rate, q the pitch rate and r the yaw rate while ω

_w

represents the angular velocity vector of F

^W

relative to F

^E

.

Based on the above notations typically employed in the aeronautics literature, the state vector is given by

x

^T

=

V

_B^T

ω

^T

Φ

^T

p

^T_E

T

= [u v w p q r φ θ ψ p

_n

p

_e

p

_h

]

^T

(2.4)

Then, the equations of motion of a rigid body aircraft when considering a stationary flat-Earth and the atmosphere at rest ar given in [49] and they have the form

˙

p

_E

= R

^T_BE

V

_a

(2.5a)

Φ = ˙ G(Φ)ω (2.5b)

V ˙

B

= −Ω

B

V

B

+ R

BE

g

0

+ F

B

m (2.5c)

˙

ω = −J

⁻¹

Ω

B

J ω + J

⁻¹

T

B

(2.5d)

where Ω

_B

is the cross-product matrix of the body axes angular rates, g denotes the

gravitational acceleration, F

B

is the net applied force on the aircraft center of gravity,

m is the vehicle mass, J represents the inertia matrix of the rigid aircraft, T

_B

is the

net torque acting about the aircraft center of gravity. The complete mathematical

model given in (2.5) can be separated into translational and rotational equations as

it will be further shown.

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2.3.2 The navigation equations

Equation (2.5a) provides the coordinates of the flight path in the inertial frame and it is known as the navigation equation. Introducing (2.1) in (2.5a) and using the above notations in order to extend the equation in terms of the vector components introduced in subsection 2.3.1, the rate of change of the translational position reads

˙

p

_n

= u cos θ cos ψ + v (− cos φ sin ψ + sin φ sin θ cos ψ )

+w (sin φ sin ψ + cos φ sin θ cos ψ) (2.6a)

˙

p

_e

= u cos θ sin ψ + v (cos φ cos ψ + sin φ sin θ sin ψ)

+w (−sinφ cos ψ + cos φ sin θ sin ψ) (2.6b)

˙

p

_h

= u sin θ − v sin φ cos θ − w cos φ cos θ (2.6c)

A more convenient form of the inertial position coordinates can be obtained from equation (2.3) when employing the velocity vector in the wind axes reference frame.

The differential equations governing the translational position are given by

˙

p

_n

= V

_a

cos θ

_w

cos ψ

_w

(2.7a)

˙

p

_e

= V

_a

cos θ

_w

sin ψ

_w

(2.7b)

˙

p

_h

= −V

_a

sin θ

_w

(2.7c)

2.3.3 The attitude equations

The attitude equation is represented by (2.5b) which provide the orientation of the airplane in the inertial frame. Note from this equation that the angular velocity vector of the aircraft in the body-fixed reference frame is related to the angular velocity vector in the inertial frame through a transformation matrix G(Φ) defined as

G(Φ) =







1 sin φ tan θ cos φ tan θ 0 cos φ − sin φ 0 sin φ sec θ cos φ sec θ







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Thus, the rate of change of angular position reads

φ ˙ = p + tan θ (q sin φ + r cos φ) (2.8a)

θ ˙ = q cos φ − r sin φ (2.8b)

ψ ˙ = q sin φ + r cos φ

cos θ (2.8c)

2.3.4 The force and moment equations

The last two equations derived in (2.5) are driven by the forces and moments acting on the aircraft center of gravity. They have both components due to several factors with main sources being the propulsive and aerodynamic effects. In order to extend the dynamic equations in terms of vector components and forces and moments acting on the aircraft, let us first examine how F

_B

and T

_B

can be expressed.

The force vector from equation (2.5c) can be represented in terms of the propulsive and aerodynamic components, F

_p

and F

_a

respectively. Thus

F

_B

= F

_p

+ F

_a

The propulsive force is produced by the engine thrust denoted by T . Generally, the engines are placed along the longitudinal body axis of the aircraft so that the produced force has just a component pointing in the direction of this axis, that is

F

p

=





 T

0 0







Regarding the aerodynamic force, it can be expressed in both body or wind axes;

whether one frame or another is employed, the components are related by the rotation

matrix given in equation (2.2). Actually, the aerodynamic force may be naturally

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defined in the wind axes as lift L, drag D and sideforce Y . Hence

F

_a^W

=







−D Y

−L







=







¯ qSC

_D

¯ qSC

Y

¯ qSC

_L







where ¯ q =

¹₂

ρ(h)V

_a²

is the free-stream dynamic pressure, S is the wing area and C

_D

, C

L

, C

Y

are dimensionless aerodynamic coefficients which are primarily dependent on aerodynamic angles, the geometry of the aircraft, the deflections of the control surfaces, etc.

Denoting the components of the aerodynamic force in the body axes by (X

_a

, Y

_a

, Z

_a

), they can be expressed in terms of body-axes dimensionless aerodynamic coefficients C

_x

, C

_y

, C

_z

F

_a

=





 X

_a

Y

_a

Z

a







=







¯ qSC

_x

¯ qSC

_y

¯ qSC

z







or in terms of the wind axes components of the aerodynamic force X

_a

= −D cos α cos β − Y cos α sin β + L sin α

Y

_a

= −D sin β + Y cos β

Z

_a

= −D sin α cos β − Y sin α sin β − L cos α

Extending the force equation (2.5c) and introducing the above notations, it yields

˙

u = rv − qw − g

₀

sin θ + X

_a

+ T

m (2.9a)

˙

v = −ru + pw + g

₀

sin φ cos θ + Y

_a

m (2.9b)

˙

w = qu − pv + g

₀

cos φ cos θ + Z

_a

m (2.9c)

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In some particular cases, e.g. derivation of a linear small-perturbation model or estimation of the aerodynamic derivatives values, it is more convenient to express the force equations of motion in the wind axes in which the aerodynamic coefficients are naturally measured. In the following, we will provide the final form of these equations, the reader being referred to [17] for a detailed description of how the equations were obtained.

The scalar expansion of the wind axes equations reads

m V ˙

_a

= T

_xw

− D − mg sin θ

_w

(2.10a)

˙

α = q − q

_w

sec β − p cos α tan β − r sin α tan β (2.10b)

β ˙ = r

_w

+ p sin α − r cos α (2.10c)

where T

_xw

represents the thrust component along the x-axis of the wind frame and [p

_w

q

_w

r

_w

] are the angular velocities of the wind frame relative to the inertial frame of reference.

A similar procedure applied to the moment equation (2.5d), in which T

B

is defined in terms of aerodynamic and propulsive components, leads to

T

_B

= T

_p

+ T

_a

=





 0 dT

0 



 +





 L ¯

_a

M

_a

N

_a







=







¯ qSbC

_l

dT + ¯ qS¯ cC

_m

¯ qSbC

_n







where d is the offset of the engine from the aircraft center of gravity along the z-axis of the body frame, T

_a^B

is the moment due to aerodynamic effects having components ( ¯ L

_a

M

_a

N

_a

) in the direction of the body axes, b represents the wing span, ¯ c defines the mean geometric chord of the wing and (C

_l

, C

_m

, C

_n

) are dimensionless coefficients primarily dependent on the aerodynamic angles.

Denoting the body axes moment components by T

B

= L M N ¯

T

in accordance

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with traditional usage, the expended set of the moment equation reads

˙

p = (c

₁

r + c

₂

p) q + c

₃

L ¯ + c

₄

N (2.11a)

˙

q = c

₅

pr − c

₆

p

²

− r

²

+ c

₇

M (2.11b)

˙

r = (c

₈

p − c

₂

r) q + c

₄

L ¯ + c

₉

n (2.11c)

where the constants c

_i

, i = 1, 9 are given by

Γc

₁

= (J

_y

− J

_z

) J

_z

− J

_xz²

, Γc

₂

= (J

_x

− J

_y

+ J

_z

) J

_xz

Γc

₃

= J

_z

, Γc

₄

= J

_xz

c

₅

=

^J^z_J^−J^x

y

, c

₆

=

^J_J^xz

y

c

₇

=

_J¹

y

, Γc

₈

= J

_x

(J

_x

− J

_y

) + J

_xz²

Γc

₉

= J

_x

, Γ = J

_x

J

_z

− J

_xz²

Remark: Writing the wind-axes moment equations offers no advantages for use in a nonlinear model. In reality, these equations are more complex than the previously derived body-axes equations. Therefore, typical nonlinear models combine force equations in either body or wind axes with body-axes moment equations [49].

2.3.5 Discussion of the equations

The complete state model of the airplane consists of 12 coupled nonlinear ordinary

differential equations obtained from (2.6), (2.8), (2.9) and (2.11). Note that two alter-

natives have been presented for both navigation and force equations given in (2.7) and

(2.10). The control vector, although it is not directly observable in these equations,

determines the thrust force and the deflections of the movable surfaces managing the

aerodynamic forces (D, L, Y ) and moments ( ¯ L, M , N ). The mathematical model

established by collecting these equations is subject to some general assumptions such

as: (i) the airplane is a rigid body having a plane of symmetry, (ii) the Earth is flat

and stationary and (iii) the atmosphere is at rest relative to the Earth.

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2.4 Flying in a moving atmosphere

It has been shown in the previous section that airplane dynamics offer challenging control problems since they are nonlinear, require transformations between several reference frames and depend on uncertain forces and moments. In addition, in real conditions airplanes are subjected to environmental disturbances such as wind which is the movement of the surrounding air that disturbs the stability of the vehicle and its inertial track. Hence, an analysis of how these perturbations affect the dynamics of flight is required to obtain improved flight capabilities.

Accordingly, the aim of this section is to extend the mathematical model previously derived by including the effects of the wind on the aircraft performance. For this reason, the section begins by describing a model of low altitude wind which is adequate for analysis purposes. Further, the vulnerability of airplanes to wind is analyzed and incorporated into the equations of motion.

2.4.1 Wind description

To understand and analyze how the air motion impacts the modeling of an aircraft, we need first to describe the wind itself as part of the velocity field in which the aircraft flies. The air mass is in a continuous state of motion due to the solar heating, Earth rotation or various thermodynamic and electromagnetic processes. The velocity vector of the atmosphere is generally variable in both space and time and it can be decomposed into a mean value and variations from it [17]. The steady-state velocity at a given position is known as mean wind while the remaining fluctuating part is defined as atmospheric turbulence or gust. The wind occurs primarily in navigation and guidance applications while the turbulence affects mainly the airplane stability.

The conventional notation for the velocity vector of the air mass relative to the Earth is W. Based on the above considerations, the total velocity field within the atmosphere is defined as

W = W

_M

+ W

_F

where W

M

is the mean wind vector and W

F

is the atmospheric turbulence.

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Local wind is naturally measured in the direction of the Earth-fixed reference frame having north, east and down velocity components denoted by W

_n

, W

_e

and W

_h

, respectively. For convenience, it may be represented in other coordinate systems using the transformation matrices derived in the previous section. Thus

W =





 w

n

w

_e

w

h







=







w

n_M

+ w

n_F

w

_e_M

+ w

_e_F

w

h_M

+ w

h_F







= R

_IB





 u

w

v

_w

w







where (u

_w

, v

_w

, w

_w

) represent the wind components in the body-fixed reference frame and R

_IB

is given by equation (2.1) with R

_IB

= R

^T_BI

.

The investigation of the wind vector effect on the flying qualities of an aircraft requires a mathematical model of such perturbation. In principle, a deterministic description of complete wind is not possible; in other words, it can not be described by analytical expressions. Rather, the wind field can be modeled as a stochastic process for which statistical properties can be described [1, 17, 50]. The derivation of a wind gust model relies heavily on the random-process theory, the reader being referred to [17, 1] for a detailed discussion on the subject.

There are two spectral forms of random continuous turbulence used to model atmospheric turbulence which were provided by the scientists von Karman and Dry- den. To generate the fluctuating wind vector with the correct characteristics, the Von Karman velocity spectra are used to filter a unit variance, band-limited white noise signal. The transfer functions of a Von Karman model are further listed [41].

H

u

(s) =

σ

_u

q

2

π Lu

V

1 + 0.25

^L_V^u

s 1 + 1.357

^L_V^u

s + 0.1987

^L_V^u

2

s

²

H

_v

(s) =

σ

v

r

1 π

Lv

V

1 + 2.7478

^L_V^v

s + 0.3398

^L_V^v

2

s

²

1 + 2.9958

^L_V^v

s + 1.9754

^L_V^v

2

s

²

+ 0.1539

^L_V^v

3

s

³

H

_w

(s) =

σ

_w

r

1 π

Lw

V

1 + 2.7478

^L_V^w

s + 0.3398

^L_V^w

2

s

²

1 + 2.9958

^L_V^w

s + 1.9754

^L_V^w

2

s

²

+ 0.1539

^L_V^w

3

s

³

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where L

_u

, L

_v

, L

_w

represent the turbulence scale lengths, σ

_u

, σ

_v

, σ

_w

represent the turbulence intensities and V is the speed of the vehicle.

In terms of Dryden models, the forming filters are derived from the spectral square roots of the spectrum equations and the transfer functions are given by [41]

H

_u

(s) = σ

_u

r 2L

_u

πV 1 1 +

^L_V^u

s H

_v

(s) = σ

_v

r L

v

πV 1 +

√ 3Lv

V

s 1 +

^L_V^v

s

2

H

_w

(s) = σ

_w

r L

_w

πV 1 +

√3Lw

V

s 1 +

^L_V^w

s

2

The parameters of a Dryden gust model are summarized in Table 2.1 for different conditions of flight and an example of a low altitude low turbulence Dryden gust model is illustrated in Figure 2 − 9 having the mean values equal to w

_n_M

= 4 m/s, w

e_M

= 1.7 m/s and w

h_M

= 0.4 m/s. In real conditions, the parameters of the mean wind change along the flight path due to the movement of air masses relative to one another. A significant variation over a relatively short distance in either the speed or direction of the wind is called wind shear and it is of special interest during the take-off and landing approaches [42].

0 5 10 15 20 25

3.5 4 4.5

w n

0 5 10 15 20 25

1 2 3

w e

0 5 10 15 20 25

0 0.5 1

Time [s]

w h

Figure 2-9: Illustration of a low altitude low turbulence Dryden gust model.

Small lightweight aircraft navigation in the presence of wind

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